On driftless one-dimensional sdes with time-dependent diffusion coefficients

Consider a one-dimensional stochastic differential equation (S) without drift but with time-dependent diffusion coefficient. To obtain weak solutions, the time change method is applied: An increasing process is looked for such that a given Brownian motion, distorted by this process in its time argum...

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Bibliographic Details
Published inStochastics and stochastics reports Vol. 67; no. 3-4; pp. 207 - 230
Main Author Raupach, Peter
Format Journal Article
LanguageEnglish
Published Abingdon Gordon and Breach Science Publishers 01.09.1999
Taylor & Francis
Subjects
AMS Classifications:60H10, 60H05, 34F05
Exact sciences and technology
Mathematical analysis
Mathematics
monotone approximation
Ordinary differential equations
Probability and statistics
Probability theory and stochastic processes
pure martingales
representability
Sciences and techniques of general use
Stochastic analysis
Stochastic differential equations
time change
time-dependent diffusion
Brownian motion
Solution uniqueness
Time dependence
Time change
Differential equation
Existence of solution
Martingale
Monotone approximation
Diffusion
Pure martingale
Stochastic equation
Stochastic differential equation
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Summary:Consider a one-dimensional stochastic differential equation (S) without drift but with time-dependent diffusion coefficient. To obtain weak solutions, the time change method is applied: An increasing process is looked for such that a given Brownian motion, distorted by this process in its time argument, turns out to be a solution of (S). For this purpose, a time change equation(TC) has to be solved. We present one-to-one relations between (S) and (TC) concerning existence and uniqueness. Contrary to SDEs, pathwise uniqueness for (TC) coincides with that in law. By solving (TC), solutions of (S) are constructed under weak conditions admitting degenerate diffusion. The results improve those of Senf [18] and Rozkosz and Słomin ński [17]. Apart from degenerate diffusion, the main difference is that monotone approximation for the solutions of (TC), in contrast to weak convergence, is systematically exploited. As a consequence, the constructed solutions can be identified as pure so that they have the representation property
ISSN:1045-1129
1029-0346
DOI:10.1080/17442509908834211